P-Bass - Very fundamental, deep, round tone. Humbucking pickup located in a place of high string excursion gives it, in the immortal words of Bass NW, 'A distinctive, punchy tone'...

J-Bass - a more customizable tone due to two single-coil pickups placed in two distinctly different areas of the body and the ability to blend them together to varying degrees... again 'A distinctive, punchy tone'...

Which one I prefer depends upon the moment... They've both been used to play everything out there at one time or another, and while I wouldn't necessarily say that one was better than the other for a certain style of music, the P does seem to be in the hands of more hard rockers...

If you're asking this question, you're probably a newbie (no offense meant, just an observation). If you are a newbie, subtleties of tone probably aren't critical to you (and besides, both P's and J's are used for hard rock).

With that in mind, here's my two cents: unless you have very small hands, I recommend the Precision simply because the Jazz bass neck is very narrow, and a lot of players who become accustomed to very narrow string spacing find it very difficult to adjust to medium or wide string spacing later on.